Restricted maximum of non-intersecting Brownian bridges

Consider a system of N non-intersecting Brownian bridges in [0,1], and let M-N(p) be the maximal height attained by the top path in the interval [0, p], p is an element of [0, 1]. It is known that, under a suitable rescaling, the distribution of M-N(p) converges, as N -> infinity, to a one-parameter family of distributions interpolating between the Tracy-Widom distributions for the Gaussian Orthogonal and Unitary Ensembles (corresponding, respectively, to p -> 1 and p -> 0). It is also known that, for fixed N, M-N(1) is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. Here we show a version of these results for M-N(p) for fixed N, showing that M-N(p) / root p converges in distribution, as p -> 0, to the rightmost charge in a generalized Laguerre Unitary Ensemble, which coincides with the top eigenvalue of a random matrix drawn from the Antisymmetric Gaussian Ensemble.